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On some convexity properties of Musielak-Orlicz spaces. (English) Zbl 0573.46015
Two kinds of rotundity, local uniform rotundity and uniform rotundity in every direction in Musielak-Orlicz spaces $$L_{\phi}$$ are examined. For the property of local uniform rotundity it is a continuation of the author’s paper devoted to Orlicz spaces, but for uniform rotundity in every direction the results are new even for Orlicz spaces.
It is proved that rotundity, local uniform rotundity and uniform rotundity in every direction are equivalent in Musielak-Orlicz spaces equipped with Luxemburg norm in the case of an atomless measure. These properties are also expressed in the terms of the function $$\phi$$.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B25 Classical Banach spaces in the general theory 46B20 Geometry and structure of normed linear spaces